Subgroups — C₃₇⋊H(3,𝔽₉) #41008044

Order 26973.

1334 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582550	C₃	1	3	yes	maximal, complemented
2	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	36	yes	maximal, complemented
3	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	9	yes	complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	72	yes	complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	36	yes	complemented
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic, complemented
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	complemented
9	C₃³×C₃₇	—	9	4	no	—
10	C₃³×C₃₇	C₃³	1	9	yes	—
11	C₃³×C₃₇	C₃³	1	4	yes	—
12	C₃²×C₃₇⋊C₃	—	9	36	no	—
13	C₃²×C₃₇⋊C₃	C₃³	1	27	yes	—
14	H(3,𝔽₃)×C₃₇	—	9	12	no	—
15	C₃²⋊(C₃₇⋊C₃)	—	9	36	no	—
16	C₃²⋊(C₃₇⋊C₃)	—	9	72	no	—
17	H(3,𝔽₉)	—	37	1	no	maximal, Sylow
18	C₃²×C₃₇	—	3	12	no	—
19	C₃²×C₃₇	—	3	36	no	—
20	C₃²×C₃₇	—	3	4	no	—
21	C₃²×C₃₇	—	9	9	no	—
22	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
23	C₃×C₃₇⋊C₃	—	3	108	no	—
24	C₃×C₃₇⋊C₃	—	9	81	no	—
25	C₃²⋊H(3,𝔽₃)	—	37	3	no	—
26	C₃²⋊H(3,𝔽₃)	—	37	36	no	—
27	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
28	C₃×C₃₇	—	9	9	no	—
29	C₃×C₃₇	—	9	4	no	—
30	C₃×C₃₇	H(5,𝔽₃)	1	4	yes	—
31	C₃₇⋊C₃	—	9	27	no	—
32	C₃⁴	—	37	9	no	—
33	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
34	C₃×H(3,𝔽₃)	—	37	36	no	—
35	C₃×H(3,𝔽₃)	—	37	72	no	—
36	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	complemented
37	C₃₇	H(3,𝔽₉)	1	1	yes	characteristic, complemented, Sylow
38	C₃³	—	9	4	no	—
39	C₃³	—	37	27	no	—
40	C₃³	—	333	36	no	—
41	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
42	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
43	H(3,𝔽₃)	—	9	12	no	—
44	H(3,𝔽₃)	—	333	72	no	—
45	H(3,𝔽₃)	—	333	36	no	—
46	C₃²	—	3	4	no	—
47	C₃²	—	3	36	no	—
48	C₃²	—	3	12	no	—
49	C₃²	—	9	9	no	—
50	C₃²	—	333	81	no	—
51	C₃²	—	111	108	no	—
52	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃	—	9	4	no	—
54	C₃	—	9	9	no	—
55	C₃	—	333	27	no	—
56	C₃	C₃₇⋊H(5,𝔽₃)	1	4	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	H(3,𝔽₉)
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊H(3,𝔽₃)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22582550	C₃	semidirect
2. (C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
3. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
4. C₃³×C₃₇⋊C₃	C₃²	semidirect
5. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
6. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
7. C₃⁴×C₃₇	C₃²	semidirect
8. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
27. C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	semidirect
33. C₃⁴	C₃×C₃₇⋊C₃	semidirect
36. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
37. C₃₇	H(3,𝔽₉)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22582550	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
H(3,𝔽₉)	37

Subgroups — C37⋊H(3,𝔽9) #41008044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊H(3,𝔽₉) #41008044